19.1.1 Observed data

19.1.2 Variability of the statistic

19.1.3 Bootstrap SE confidence interval

Once we have a bootstrapped sampling distribution, we can use it to make confidence intervals.

• The 95% percentile confidence interval is the the interval from the 2.5%ile to the 97.5%ile.
• The 95% SE confidence interval is the quick approximation \(\mbox{point estimate} \pm 2 \cdot SE_{BS}\)
  • We are using 2 instead of 1.96.

19.1.5 Bootstrapping is not a solution to small sample sizes!

• Note: Bootstrapping (and other statistical models) work best for larger samples.
• The car price example had a sample size of \(n=5\), which is pretty small.

19.2.1 Mathematical distribution of the sample mean

Central Limit Theorem for the sample mean. When we collect a sufficiently large sample of \(n\) independent observations from a population with mean \(\mu\) and standard deviation \(\sigma\), the sampling distribution of \(\bar{x}\) will be nearly normal with mean \(\mu\) and standard deviation \(SE = \sigma/\sqrt{n}\).

• In practice, we don’t usually know \(\sigma\), so we use \(SE \approx s/\sqrt{n}\).

19.2.2 Evaluating the two conditions required for modeling \(\bar{x}\)

• Independence. e.g., the sample is a simple random sample from the population.
• Normality. If the sample is small, then the data should come from a normal, or nearly normal, population.
  • “small” means less than 15
  • “large” means greater than 60
  • Extreme outliers can cause problems.

19.2.3 Introducing the t-distribution

• For smaller samples, the sampling distribution of \(\bar{x}\) has thicker tails than the normal distribution.
• A better approximation is the \(t-distribution\).
  • As with the chi-squared distribution, the t-distribution requires you to specify the degrees of freedom.
  • The sampling distribution of \(\bar{x}\) has \(n-1\) degrees of freedom, where \(n\) is the sample size.

Tail probabilities of the t-distribution in R

As with pnorm and pchisq, you can get lower tail probabilities in R using pt.

pt(-2.5, df = 19)  # lower tail below -2.5

[1] 0.01087021

1 - pt(2.5, df = 19) # upper tail above 2.5

[1] 0.01087021

Quantiles of the t-distribution in R

If you know the tail probability you want, you can get the corresponding t.

qt(0.025, df = 11)

[1] -2.200985

qt(0.975, df = 11)

[1] 2.200985

Recall that for the normal distribution, these values were \(\pm 1.96\) (fatter tails).

19.2.4 One sample t-intervals

The generic formulas for confidence intervals work for t-intervals:

\[ \begin{aligned} \text{point estimate} \ &\pm\ t^{\star}_{df} \times SE \\ \bar{x} \ &\pm\ t^{\star}_{df} \times \frac{s}{\sqrt{n}} \end{aligned} \]

To get \(t^\star_{df}\) you can use qt in R and you need the degrees of freedom (\(n-1\)).

19.2.5 One sample t-tests

\[ H_0: \mu = \text{null value} \\ H_A: \mu \neq \text{null value} \]

The generic \(Z\)-score formula gives us a formula for \(T\)-score:

\[ T = \frac{\bar{x} - \mbox{null value}}{s/\sqrt{n}} \]

t-tests in R

The frogs vector contains the weights of the 14 frogs in our sample.

t.test(frogs, mu = 24)

    One Sample t-test

data:  frogs
t = 2.3008, df = 13, p-value = 0.0386
alternative hypothesis: true mean is not equal to 24
95 percent confidence interval:
 24.14950 28.74841
sample estimates:
mean of x 
 26.44895